The identity-assignment planar measure partition conjecture

Let $k$ be a positive integer, let $\mu_1,\dots,\mu_k$ be absolutely continuous probability measures in the plane, and let $\alpha_1,\dots,\alpha_k$ be positive reals with sum $1$.

Identity-assignment partition conjecture. There exists a positive integer $r$ such that the plane can be partitioned, using $r$ lines, into $2k$ not necessarily connected regions $Q_1,\dots,Q_k$ satisfying

$$\mu_i(Q_i)=\alpha_i\qquad\text{for all }i\in[k].$$

The conjecture is intended to generalize a one-dimensional result and is presented without a resolution in the supplied context.